String Compactification, Effective Field Theory And Holography Swampland

Abstract: This thesis primarily dives into investigating the details of four-dimensional vacua within String Theory using the AdS/CFT correspondence. In the first and second part of the thesis, we study the fibred Calabi-Yau and M-theory moduli stabilization scenario. We consider both flux-stabilized models and non-perturbative stabilization methods. We perform a holographic analysis to determine the spectrum of the assumed dual $CFT_3$ to understand its AdS/CFT implications. For the flux stabilization, which relies on a large complex Chern-Simons invariant, moduli have integer dimensions similar to the DGKT flux-stabilized model in type IIA. For the non-perturbative stabilization, the results are similar to race-track models in type IIB. In the last part of the thesis, we solve the Wheeler DeWitt equation for the planar Reissner-Nordstrom-AdS black hole in a minisuperspace approximation. We construct semiclassical Wheeler-DeWitt states from Gaussian wavepackets that are peaked on classical black hole interior solutions. By using the metric component $g_{xx}$ as a clock, these states are evolved through both the exterior and interior horizons. Close to the singularity, we show that quantum fluctuations in the wavepacket become important, and therefore the classicality of the minisuperspace approximation breaks down. Towards the AdS boundary, the Wheeler-DeWitt states are used to recover the Lorentzian partition function of the dual theory living on this boundary. This partition function is specified by an energy and a charge. Finally, we show that the Wheeler-DeWitt states know about the black hole thermodynamics, recovering the grand canonical thermodynamic potential after an appropriate averaging at the black hole horizon.

• The paper demonstrates that explicit moduli stabilization in Calabi-Yau and M-theory settings yields detailed dual CFT spectra that challenge traditional swampland sign conjectures.
• It employs rigorous top‐down analyses contrasting flux and nonperturbative stabilization, utilizing effective field theory techniques to derive conformal dimensions and operator data.
• The study integrates Wheeler-DeWitt quantization for RN-AdS black holes, revealing quantum gravitational effects that bridge holographic renormalization group flows with black hole thermodynamics.

String Compactification, Holographic Swampland, and Quantum Black Holes: An Expert Analysis

Moduli Stabilization, Swampland Constraints, and Holography

The dissertation systematically investigates 4d string theory vacua in the context of the AdS/CFT correspondence, centering on the interplay between string/field-theoretic consistency (Swampland conditions), moduli stabilization mechanisms, and the holographic spectra of conjectured dual CFTs. The primary focus is twofold: (1) examining top-down moduli stabilization scenarios for fibred Calabi-Yau and M-theory compactifications (both flux and nonperturbatively stabilized), and (2) leveraging quantum cosmological methods—specifically, the construction and propagation of Wheeler-DeWitt (WdW) states—for quantum Reissner-Nordström-AdS black holes, with attention to their dual interpretation.

A foundational aspect is the detailed moduli stabilization analysis across several string-derived effective field theory (EFT) settings. The work expounds on both IIB and M-theory scenarios, utilizing fluxes and nonperturbative contributions to tackle the moduli problem. Holographic duals are then constructed for these vacua, focusing on the spectrum of resulting 3d CFT operators, including their conformal dimensions and operator product expansion (OPE) data.

The Swampland program is brought into dialogue with holography by testing refined conjectures—e.g., the mixing anomalous dimension sign criterion for double-trace operators—against explicit stabilized vacua. The comparative structure is leveraged further by contrasting flux-stabilized and nonperturbative M-theory vacua, drawing explicit parallels to the DGKT (type IIA) and racetrack/KKLT (type IIB) constructions, respectively.

Fibred Calabi-Yau Scenarios and Holographic Data

The analysis of fibred Calabi-Yau LVS models is technically sophisticated, examining cases with both single and multiple fibre moduli. The author employs explicit volume forms (e.g., $$\mathcal{V} = \alpha \sqrt{\tau_1 \tau_2} - \gamma \tau_3^{3/2}$$) and systematically derives the kinetic and potential sectors, ultimately transforming to canonical coordinates for a transparent holographic analysis.

A salient result concerns the spectrum of light fields: for a single fibre modulus, there is a mass hierarchy between the volume (massive) and fibre/axion directions (massless at tree level). This leads to degeneracies in the low-energy spectrum and manifests as operator degeneracies in the dual CFT. The OPE coefficients for various couplings are computed (following the standard AdS/CFT Witten diagram prescription), and the structure of double-trace anomalous dimensions is analyzed—including the effect of discrete symmetries (e.g., $\mathbb{Z}_2$ factors) on operator mixing.

Crucially, the study finds that double-trace operators composed of distinct axions (e.g., $[a_1 a_2]_{n,\ell}$) can possess positive anomalous dimensions, in contradiction to the strict sign conjecture advanced in prior work. The implication is that, for models with multiple axions and specific moduli-mixing structure, the sign conjecture requires revision; the landscape/swampland distinction necessitates more nuanced criteria. The results are generalized to examples with two fibre moduli, examining the enlarged degeneracy structure and symmetry breaking patterns.

M-Theory Compactification: Flux and Nonperturbative Branches

The thesis investigates both flux and nonperturbative stabilization mechanisms for M-theory on $G_2$-holonomy manifolds. The flux-stabilized branch (linear superpotential) yields highly nontrivial spectra:

• The moduli and axion conformal dimensions in the dual CFT are integer-valued; specifically, for $N$ moduli, $\Delta_{1,\cdots,N-1}=2$, $\Delta_N=8$ for saxions, and $\Delta_{1,\cdots,N-1}=3$, $\Delta_N=7$ for axions.
• This integer spectrum is highly reminiscent of the spectrum found in type IIA DGKT flux vacua, a fact emphasized by cross-comparison of the Kähler and superpotential structures.

In the nonperturbative branch (racetrack-like superpotential with multiple exponential terms), the analysis shows that most conformal dimensions remain fixed (as above), but one operator acquires a conformal dimension scaling as $\Delta \sim \log^2 R_\text{AdS}$ in the scale-separated limit. This “heavy operator” behavior aligns with expectations from racetrack and generalizes the KKLT result, where $\Delta \sim \log R_\text{AdS}$.

The explicit formulae for the Hessian and conformal data in both branches are provided, linking the Swampland/landscape distinction at the level of the dual CFT spectrum. The theoretical challenge posed by the emergence of integer conformal dimensions is highlighted as an open direction requiring deeper understanding—potentially of a number-theoretic or geometric origin.

Wheeler-DeWitt Quantization and Holographic Flow for Charged Black Holes

The final segment turns to quantum gravitational dynamics: the WdW equation is semiclassically solved for the planar RN-AdS black hole in a minisuperspace approximation, with $g_{xx}$ adopted as an internal “clock” due to non-monotonicity of $g_{tt}$ in charged backgrounds. Semiclassical WdW states are constructed as Gaussian superpositions peaked on classical solutions and evolved across both exterior and interior horizons.

Key technical results include:

• The variance of $g_{tt}$ diverges near the timelike singularity in RN-AdS, signaling a breakdown of the minisuperspace classicality; quantum fluctuations become dominant, in contrast to the spacelike singularity case.
• At the AdS boundary, the Wheeler-DeWitt states reduce to the Lorentzian partition function of the dual field theory (parameterized by energy and charge), providing an explicit realization of the holographic renormalization group flow in WdW language.
• Upon appropriate averaging (partially integrating out metric data at the horizon), the grand canonical thermodynamic potential of the black hole is exactly recovered—demonstrating that quantum gravitational WdW states encode precise thermodynamic information accessible holographically.

Implications and Future Directions

The results have several important theoretical implications:

• The observation of integer conformal dimensions in dual CFTs for flux-stabilized vacua (M-theory and DGKT) calls for an explanation within the AdS/CFT framework and may imply deeper algebraic or geometric constraints on allowed string compactifications.
• The breakdown of the sign conjecture for axion-axion double-trace operators underlines the complexity of the swampland/landscape boundary, encouraging refined or context-dependent swampland criteria.
• The quantum WdW construction for black holes provides a blueprint for tracing RG flow through horizons and singularities in a fully covariant manner, with applications to black hole information, cosmic censorship, and the resolution of singularities.
• The connection between WdW states and boundary operator flows points toward a richer interplay between quantum gravity constraints and CFT data (including possible relations to $T\bar{T}$-like deformations).

Going forward, three concrete directions are suggested: elucidation of the number-theoretic origin of integer conformal dimensions, deeper investigation of $G_2$ manifolds with large Chern-Simons invariants, and the extension of WdW formalism to include matter or deformations beyond minisuperspace. Each of these carries significant implications for both the theoretical structure of string theory and the practical realization of Swampland/holographic constraints.

Conclusion

This thesis advances the interface between string compactification, Swampland conjectures, and holography by combining explicit EFT computations, detailed canonical analyses, and rigorous CFT interpretation. The work provides comprehensive evidence that holographic and Swampland consistency criteria must be revisited in complex models with multiple axions/fibered moduli, reveals deep structure in flux-induced CFT spectra, and constructs a robust quantum-gravitational perspective via the Wheeler-DeWitt approach. These results offer new tools and conceptual bridges for quantum gravity, holography, and the study of consistent string vacua.